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De Finetti's Theorem: Let $(X_i)_{i\in\mathbb{N}}$ be a family of exchangeable random elements defined on a measurable space $(\mathsf{X},\mathcal{X})$. Then, there exists a $\sigma$-field $\mathcal{G}_\infty$ such that, conditionally on $\mathcal{G}_\infty$, the random variables $(X_i)_{i\in\mathbb{N}}$ are independent and identically distributed (i.i.d.).
The proof is based on the paper “Uses of exchangeability” by J. F. Kingman Click here to see the paper.
Without loss of generality, we model $(X_i)_{i\in\mathbb{N}}$ as the coordinate projections on the canonical probability space $(\mathsf{X}^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}},\mathbb{P})$. We proceed as follows:
1. Reverse filtration construction:
$$ f(x_1,\ldots,x_n,x_{n+1},\ldots)=f(x_{\pi(1)},\ldots,x_{\pi(n)},x_{n+1},\ldots). $$
$$ \mathcal{G}_\infty=\bigcap_{n\in\mathbb{N}}\mathcal{G}_n. $$
2. Conditional expectation and empirical averages:
$$ \mathbb{E}\!\left[\left(\frac1n\sum_{i=1}^n h(X_i)\right)\mathbf1_A\right] =\frac1n\sum_{i=1}^n \mathbb{E}\!\left[\left( h(X_i)\right)\mathbf1_A\right] =\mathbb{E}[h(X_1)\mathbf1_A]. $$
$$ \frac1n\sum_{i=1}^n h(X_i)=\mathbb{E}[h(X_1)\mid\mathcal{G}_n], \quad a.s. $$
$$ \frac1n\sum_{i=1}^n h(X_i)\xrightarrow{\mathrm{a.s.}}\mathbb{E}[h(X_1)\mid\mathcal{G}_\infty]. $$
3. Multivariate functions:
$$ \mathbb{E}[f(X_1,\ldots,X_k)\mid\mathcal{G}_\infty] =\lim_{n\to\infty}\frac1{n(n-1)\cdots(n-k+1)} \sum_{\substack{i_{1:k} \in [1:n]^k \le n,\ldots,1 \le i_k\le n\\ i_j\neq i_\ell}} f(X_{i_1},\ldots,X_{i_k}). $$
$$ \frac1{n(n-1)\cdots(n-k+1)} \sum_{\substack{1\le i_1,\ldots,i_k\le n\\ i_j\neq i_\ell}} f(X_{i_1},\ldots,X_{i_k}) + O\lr{\frac1n} = \frac1{n^k}\sum_{i_1=1}^n\cdots\sum_{i_k=1}^n f(X_{i_1},\ldots,X_{i_k}). $$
$$ \mathbb{E}[f_1(X_1)\cdots f_k(X_k)\mid\mathcal{G}_\infty] =\prod_{\ell=1}^k\mathbb{E}[f_\ell(X_1)\mid\mathcal{G}_\infty]. $$